The performance of nonblocking input-output-buffered ATM (asynchronous transfer mode) switch architectures with backpressure control is investigated, when operating under different selection mechanisms. The unfairness problem associated with the classical cyclic service mechanism in broadband switches with input and output buffering and back-pressure control is explained and three alternative service or selection mechanisms are proposed. The proposed selection policies are compared from the point of view of fairness, implementation complexity, and cell loss performance. The different selection policies have a strong impact on the (complementary) input queue length distributions and the cell loss behavior of the switch. It is found that the queue length criterion results in a significantly better performance in terms of queue length distribution and cell loss behavior, and is easier to implement than other selection criteria that involve time stamping.